LMFDB - Embedded newform 1200.3.c.b.449.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,3,Mod(449,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1200.449"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-18,0,0,0,0,0,0,0,0,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			449.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.b.449.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +13.0000i q^{7} -9.00000 q^{9} -23.0000i q^{13} +11.0000 q^{19} +39.0000 q^{21} +27.0000i q^{27} -59.0000 q^{31} +26.0000i q^{37} -69.0000 q^{39} +83.0000i q^{43} -120.000 q^{49} -33.0000i q^{57} -121.000 q^{61} -117.000i q^{63} +13.0000i q^{67} +46.0000i q^{73} -142.000 q^{79} +81.0000 q^{81} +299.000 q^{91} +177.000i q^{93} +167.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} + 22 q^{19} + 78 q^{21} - 118 q^{31} - 138 q^{39} - 240 q^{49} - 242 q^{61} - 284 q^{79} + 162 q^{81} + 598 q^{91}+O(q^{100}) $ 2 * q - 18 * q^9 + 22 * q^19 + 78 * q^21 - 118 * q^31 - 138 * q^39 - 240 * q^49 - 242 * q^61 - 284 * q^79 + 162 * q^81 + 598 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 3.00000i	− 1.00000i
$3$	− 3.00000i	− 1.00000i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	13.0000i	1.85714i	0.371154	+	0.928571i	$0.378962\pi$
$7$	13.0000i	1.85714i	−0.371154	+	0.928571i	$0.621038\pi$
$8$	0	0
$9$	−9.00000	−1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 23.0000i	− 1.76923i	−0.466321	−	0.884615i	$-0.654421\pi$
$13$	− 23.0000i	− 1.76923i	0.466321	−	0.884615i	$-0.345579\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	11.0000	0.578947	0.289474	−	0.957186i	$-0.406520\pi$
$19$	11.0000	0.578947	0.289474	+	0.957186i	$0.406520\pi$
$20$	0	0
$21$	39.0000	1.85714
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−59.0000	−1.90323	−0.951613	−	0.307299i	$-0.900575\pi$
$31$	−59.0000	−1.90323	−0.951613	+	0.307299i	$0.900575\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	26.0000i	0.702703i	0.936244	+	0.351351i	$0.114278\pi$
$37$	26.0000i	0.702703i	−0.936244	+	0.351351i	$0.885722\pi$
$38$	0	0
$39$	−69.0000	−1.76923
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	83.0000i	1.93023i	0.261822	+	0.965116i	$0.415677\pi$
$43$	83.0000i	1.93023i	−0.261822	+	0.965116i	$0.584323\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−120.000	−2.44898
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 33.0000i	− 0.578947i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−121.000	−1.98361	−0.991803	−	0.127774i	$-0.959217\pi$
$61$	−121.000	−1.98361	−0.991803	+	0.127774i	$0.959217\pi$
$62$	0	0
$63$	− 117.000i	− 1.85714i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.0000i	0.194030i	0.995283	+	0.0970149i	$0.0309295\pi$
$67$	13.0000i	0.194030i	−0.995283	+	0.0970149i	$0.969071\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	46.0000i	0.630137i	0.949069	+	0.315068i	$0.102027\pi$
$73$	46.0000i	0.630137i	−0.949069	+	0.315068i	$0.897973\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−142.000	−1.79747	−0.898734	−	0.438494i	$-0.855512\pi$
$79$	−142.000	−1.79747	−0.898734	+	0.438494i	$0.855512\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	299.000	3.28571
$92$	0	0
$93$	177.000i	1.90323i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	167.000i	1.72165i	0.508902	+	0.860825i	$0.330052\pi$
$97$	167.000i	1.72165i	−0.508902	+	0.860825i	$0.669948\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	194.000i	1.88350i	0.336321	+	0.941748i	$0.390817\pi$
$103$	194.000i	1.88350i	−0.336321	+	0.941748i	$0.609183\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−71.0000	−0.651376	−0.325688	−	0.945477i	$-0.605596\pi$
$109$	−71.0000	−0.651376	−0.325688	+	0.945477i	$0.605596\pi$
$110$	0	0
$111$	78.0000	0.702703
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	207.000i	1.76923i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 146.000i	− 1.14961i	−0.818292	−	0.574803i	$-0.805079\pi$
$127$	− 146.000i	− 1.14961i	0.818292	−	0.574803i	$-0.194921\pi$
$128$	0	0
$129$	249.000	1.93023
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	143.000i	1.07519i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−22.0000	−0.158273	−0.0791367	−	0.996864i	$-0.525216\pi$
$139$	−22.0000	−0.158273	−0.0791367	+	0.996864i	$0.525216\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	360.000i	2.44898i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−59.0000	−0.390728	−0.195364	−	0.980731i	$-0.562589\pi$
$151$	−59.0000	−0.390728	−0.195364	+	0.980731i	$0.562589\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 193.000i	− 1.22930i	−0.788800	−	0.614650i	$-0.789297\pi$
$157$	− 193.000i	− 1.22930i	0.788800	−	0.614650i	$-0.210703\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 37.0000i	− 0.226994i	−0.993538	−	0.113497i	$-0.963795\pi$
$163$	− 37.0000i	− 0.226994i	0.993538	−	0.113497i	$-0.0362052\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−360.000	−2.13018
$170$	0	0
$171$	−99.0000	−0.578947
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−1.00000	−0.00552486	−0.00276243	−	0.999996i	$-0.500879\pi$
$181$	−1.00000	−0.00552486	−0.00276243	+	0.999996i	$0.500879\pi$
$182$	0	0
$183$	363.000i	1.98361i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−351.000	−1.85714
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	− 143.000i	− 0.740933i	−0.928846	−	0.370466i	$-0.879198\pi$
$193$	− 143.000i	− 0.740933i	0.928846	−	0.370466i	$-0.120802\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−109.000	−0.547739	−0.273869	−	0.961767i	$-0.588304\pi$
$199$	−109.000	−0.547739	−0.273869	+	0.961767i	$0.588304\pi$
$200$	0	0
$201$	39.0000	0.194030
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−419.000	−1.98578	−0.992891	−	0.119027i	$-0.962022\pi$
$211$	−419.000	−1.98578	−0.992891	+	0.119027i	$0.962022\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 767.000i	− 3.53456i
$218$	0	0
$219$	138.000	0.630137
$220$	0	0
$221$	0	0
$222$	0	0
$223$	83.0000i	0.372197i	0.982531	+	0.186099i	$0.0595844\pi$
$223$	83.0000i	0.372197i	−0.982531	+	0.186099i	$0.940416\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	409.000	1.78603	0.893013	−	0.450031i	$-0.148587\pi$
$229$	409.000	1.78603	0.893013	+	0.450031i	$0.148587\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	426.000i	1.79747i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	479.000	1.98755	0.993776	−	0.111397i	$-0.0355327\pi$
$241$	479.000	1.98755	0.993776	+	0.111397i	$0.0355327\pi$
$242$	0	0
$243$	− 243.000i	− 1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 253.000i	− 1.02429i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−338.000	−1.30502
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−242.000	−0.892989	−0.446494	−	0.894786i	$-0.647328\pi$
$271$	−242.000	−0.892989	−0.446494	+	0.894786i	$0.647328\pi$
$272$	0	0
$273$	− 897.000i	− 3.28571i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	407.000i	1.46931i	0.678439	+	0.734657i	$0.262657\pi$
$277$	407.000i	1.46931i	−0.678439	+	0.734657i	$0.737343\pi$
$278$	0	0
$279$	531.000	1.90323
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 517.000i	− 1.82686i	−0.407001	−	0.913428i	$-0.633426\pi$
$283$	− 517.000i	− 1.82686i	0.407001	−	0.913428i	$-0.366574\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−289.000	−1.00000
$290$	0	0
$291$	501.000	1.72165
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−1079.00	−3.58472
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	253.000i	0.824104i	0.911160	+	0.412052i	$0.135188\pi$
$307$	253.000i	0.824104i	−0.911160	+	0.412052i	$0.864812\pi$
$308$	0	0
$309$	582.000	1.88350
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	457.000i	1.46006i	0.683413	+	0.730032i	$0.260495\pi$
$313$	457.000i	1.46006i	−0.683413	+	0.730032i	$0.739505\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	213.000i	0.651376i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−362.000	−1.09366	−0.546828	−	0.837245i	$-0.684165\pi$
$331$	−362.000	−1.09366	−0.546828	+	0.837245i	$0.684165\pi$
$332$	0	0
$333$	− 234.000i	− 0.702703i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	167.000i	0.495549i	0.968818	+	0.247774i	$0.0796992\pi$
$337$	167.000i	0.495549i	−0.968818	+	0.247774i	$0.920301\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 923.000i	− 2.69096i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	502.000	1.43840	0.719198	−	0.694805i	$-0.244510\pi$
$349$	502.000	1.43840	0.719198	+	0.694805i	$0.244510\pi$
$350$	0	0
$351$	621.000	1.76923
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−240.000	−0.664820
$362$	0	0
$363$	− 363.000i	− 1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 227.000i	− 0.618529i	−0.950976	−	0.309264i	$-0.899917\pi$
$367$	− 227.000i	− 0.618529i	0.950976	−	0.309264i	$-0.100083\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	577.000i	1.54692i	0.633847	+	0.773458i	$0.281475\pi$
$373$	577.000i	1.54692i	−0.633847	+	0.773458i	$0.718525\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	611.000	1.61214	0.806069	−	0.591822i	$-0.201591\pi$
$379$	611.000	1.61214	0.806069	+	0.591822i	$0.201591\pi$
$380$	0	0
$381$	−438.000	−1.14961
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 747.000i	− 1.93023i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 793.000i	− 1.99748i	−0.0501728	−	0.998741i	$-0.515977\pi$
$397$	− 793.000i	− 1.99748i	0.0501728	−	0.998741i	$-0.484023\pi$
$398$	0	0
$399$	429.000	1.07519
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	1357.00i	3.36725i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	769.000	1.88020	0.940098	−	0.340905i	$-0.110733\pi$
$409$	769.000	1.88020	0.940098	+	0.340905i	$0.110733\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	66.0000i	0.158273i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−358.000	−0.850356	−0.425178	−	0.905110i	$-0.639789\pi$
$421$	−358.000	−0.850356	−0.425178	+	0.905110i	$0.639789\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1573.00i	− 3.68384i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	− 503.000i	− 1.16166i	−0.814024	−	0.580831i	$-0.802728\pi$
$433$	− 503.000i	− 1.16166i	0.814024	−	0.580831i	$-0.197272\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−709.000	−1.61503	−0.807517	−	0.589844i	$-0.799189\pi$
$439$	−709.000	−1.61503	−0.807517	+	0.589844i	$0.799189\pi$
$440$	0	0
$441$	1080.00	2.44898
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	177.000i	0.390728i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 814.000i	− 1.78118i	−0.454805	−	0.890591i	$-0.650291\pi$
$457$	− 814.000i	− 1.78118i	0.454805	−	0.890591i	$-0.349709\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 526.000i	− 1.13607i	−0.823005	−	0.568035i	$-0.807704\pi$
$463$	− 526.000i	− 1.13607i	0.823005	−	0.568035i	$-0.192296\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−169.000	−0.360341
$470$	0	0
$471$	−579.000	−1.22930
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	598.000	1.24324
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	613.000i	1.25873i	0.777111	+	0.629363i	$0.216684\pi$
$487$	613.000i	1.25873i	−0.777111	+	0.629363i	$0.783316\pi$
$488$	0	0
$489$	−111.000	−0.226994
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	851.000	1.70541	0.852705	−	0.522392i	$-0.174960\pi$
$499$	851.000	1.70541	0.852705	+	0.522392i	$0.174960\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1080.00i	2.13018i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−598.000	−1.17025
$512$	0	0
$513$	297.000i	0.578947i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	803.000i	1.53537i	0.640826	+	0.767686i	$0.278592\pi$
$523$	803.000i	1.53537i	−0.640826	+	0.767686i	$0.721408\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−241.000	−0.445471	−0.222736	−	0.974879i	$-0.571499\pi$
$541$	−241.000	−0.445471	−0.222736	+	0.974879i	$0.571499\pi$
$542$	0	0
$543$	3.00000i	0.00552486i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 506.000i	− 0.925046i	−0.886607	−	0.462523i	$-0.846944\pi$
$547$	− 506.000i	− 0.925046i	0.886607	−	0.462523i	$-0.153056\pi$
$548$	0	0
$549$	1089.00	1.98361
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 1846.00i	− 3.33816i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	1909.00	3.41503
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1053.00i	1.85714i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	181.000	0.316988	0.158494	−	0.987360i	$-0.449336\pi$
$571$	181.000	0.316988	0.158494	+	0.987360i	$0.449336\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1033.00i	− 1.79029i	−0.445770	−	0.895147i	$-0.647070\pi$
$577$	− 1033.00i	− 1.79029i	0.445770	−	0.895147i	$-0.352930\pi$
$578$	0	0
$579$	−429.000	−0.740933
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−649.000	−1.10187
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	327.000i	0.547739i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	1199.00	1.99501	0.997504	−	0.0706077i	$-0.0224939\pi$
$601$	1199.00	1.99501	0.997504	+	0.0706077i	$0.0224939\pi$
$602$	0	0
$603$	− 117.000i	− 0.194030i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	814.000i	1.34102i	0.741900	+	0.670511i	$0.233925\pi$
$607$	814.000i	1.34102i	−0.741900	+	0.670511i	$0.766075\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1126.00i	1.83687i	0.395574	+	0.918434i	$0.370546\pi$
$613$	1126.00i	1.83687i	−0.395574	+	0.918434i	$0.629454\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−949.000	−1.53312	−0.766559	−	0.642174i	$-0.778033\pi$
$619$	−949.000	−1.53312	−0.766559	+	0.642174i	$0.778033\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1261.00	1.99842	0.999208	−	0.0398015i	$-0.0126726\pi$
$631$	1261.00	1.99842	0.999208	+	0.0398015i	$0.0126726\pi$
$632$	0	0
$633$	1257.00i	1.98578i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2760.00i	4.33281i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	314.000i	0.488336i	0.969733	+	0.244168i	$0.0785148\pi$
$643$	314.000i	0.488336i	−0.969733	+	0.244168i	$0.921485\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2301.00	−3.53456
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 414.000i	− 0.630137i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	122.000	0.184569	0.0922844	−	0.995733i	$-0.470583\pi$
$661$	122.000	0.184569	0.0922844	+	0.995733i	$0.470583\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	249.000	0.372197
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 1154.00i	− 1.71471i	−0.514725	−	0.857355i	$-0.672106\pi$
$673$	− 1154.00i	− 1.71471i	0.514725	−	0.857355i	$-0.327894\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−2171.00	−3.19735
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1227.00i	− 1.78603i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1318.00	1.90738	0.953690	−	0.300790i	$-0.0972504\pi$
$691$	1318.00	1.90738	0.953690	+	0.300790i	$0.0972504\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	286.000i	0.406828i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1391.00	−1.96192	−0.980959	−	0.194214i	$-0.937784\pi$
$709$	−1391.00	−1.96192	−0.980959	+	0.194214i	$0.937784\pi$
$710$	0	0
$711$	1278.00	1.79747
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−2522.00	−3.49792
$722$	0	0
$723$	− 1437.00i	− 1.98755i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 947.000i	− 1.30261i	−0.758815	−	0.651307i	$-0.774221\pi$
$727$	− 947.000i	− 1.30261i	0.758815	−	0.651307i	$-0.225779\pi$
$728$	0	0
$729$	−729.000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 1034.00i	− 1.41064i	−0.708888	−	0.705321i	$-0.750803\pi$
$733$	− 1034.00i	− 1.41064i	0.708888	−	0.705321i	$-0.249197\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−1222.00	−1.65359	−0.826793	−	0.562506i	$-0.809837\pi$
$739$	−1222.00	−1.65359	−0.826793	+	0.562506i	$0.809837\pi$
$740$	0	0
$741$	−759.000	−1.02429
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1202.00	−1.60053	−0.800266	−	0.599645i	$-0.795309\pi$
$751$	−1202.00	−1.60053	−0.800266	+	0.599645i	$0.795309\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 673.000i	− 0.889036i	−0.895770	−	0.444518i	$-0.853375\pi$
$757$	− 673.000i	− 0.889036i	0.895770	−	0.444518i	$-0.146625\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 923.000i	− 1.20970i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−671.000	−0.872562	−0.436281	−	0.899811i	$-0.643705\pi$
$769$	−671.000	−0.872562	−0.436281	+	0.899811i	$0.643705\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1014.00i	1.30502i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	613.000i	0.778907i	0.921046	+	0.389454i	$0.127336\pi$
$787$	613.000i	0.778907i	−0.921046	+	0.389454i	$0.872664\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2783.00i	3.50946i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	1261.00	1.55487	0.777435	−	0.628963i	$-0.216520\pi$
$811$	1261.00	1.55487	0.777435	+	0.628963i	$0.216520\pi$
$812$	0	0
$813$	726.000i	0.892989i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	913.000i	1.11750i
$818$	0	0
$819$	−2691.00	−3.28571
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	563.000i	0.684083i	0.939685	+	0.342041i	$0.111118\pi$
$823$	563.000i	0.684083i	−0.939685	+	0.342041i	$0.888882\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−458.000	−0.552473	−0.276236	−	0.961090i	$-0.589087\pi$
$829$	−458.000	−0.552473	−0.276236	+	0.961090i	$0.589087\pi$
$830$	0	0
$831$	1221.00	1.46931
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1593.00i	− 1.90323i
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1573.00i	1.85714i
$848$	0	0
$849$	−1551.00	−1.82686
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1177.00i	1.37984i	0.723888	+	0.689918i	$0.242353\pi$
$853$	1177.00i	1.37984i	−0.723888	+	0.689918i	$0.757647\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	1418.00	1.65076	0.825378	−	0.564580i	$-0.190962\pi$
$859$	1418.00	1.65076	0.825378	+	0.564580i	$0.190962\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	867.000i	1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	299.000	0.343284
$872$	0	0
$873$	− 1503.00i	− 1.72165i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1727.00i	1.96921i	0.174785	+	0.984607i	$0.444077\pi$
$877$	1727.00i	1.96921i	−0.174785	+	0.984607i	$0.555923\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	443.000i	0.501699i	0.968026	+	0.250849i	$0.0807099\pi$
$883$	443.000i	0.501699i	−0.968026	+	0.250849i	$0.919290\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	1898.00	2.13498
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	3237.00i	3.58472i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	214.000i	0.235943i	0.993017	+	0.117971i	$0.0376391\pi$
$907$	214.000i	0.235943i	−0.993017	+	0.117971i	$0.962361\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	971.000	1.05658	0.528292	−	0.849063i	$-0.322833\pi$
$919$	971.000	1.05658	0.528292	+	0.849063i	$0.322833\pi$
$920$	0	0
$921$	759.000	0.824104
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1746.00i	− 1.88350i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−1320.00	−1.41783
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1847.00i	1.97118i	0.169138	+	0.985592i	$0.445902\pi$
$937$	1847.00i	1.97118i	−0.169138	+	0.985592i	$0.554098\pi$
$938$	0	0
$939$	1371.00	1.46006
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	1058.00	1.11486
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2520.00	2.62227
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1534.00i	1.58635i	0.608994	+	0.793175i	$0.291573\pi$
$967$	1534.00i	1.58635i	−0.608994	+	0.793175i	$0.708427\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	− 286.000i	− 0.293936i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	639.000	0.651376
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1739.00	−1.75479	−0.877397	−	0.479766i	$-0.840722\pi$
$991$	−1739.00	−1.75479	−0.877397	+	0.479766i	$0.840722\pi$
$992$	0	0
$993$	1086.00i	1.09366i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1894.00i	− 1.89970i	−0.312707	−	0.949850i	$-0.601236\pi$
$997$	− 1894.00i	− 1.89970i	0.312707	−	0.949850i	$-0.398764\pi$
$998$	0	0
$999$	−702.000	−0.702703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.b.449.1		2
3.2	odd	2	CM	1200.3.c.b.449.1		2
4.3	odd	2		300.3.b.b.149.2		2
5.2	odd	4		1200.3.l.a.401.1		1
5.3	odd	4		1200.3.l.e.401.1		1
5.4	even	2	inner	1200.3.c.b.449.2		2
12.11	even	2		300.3.b.b.149.2		2
15.2	even	4		1200.3.l.a.401.1		1
15.8	even	4		1200.3.l.e.401.1		1
15.14	odd	2	inner	1200.3.c.b.449.2		2
20.3	even	4		300.3.g.a.101.1	✓	1
20.7	even	4		300.3.g.c.101.1	yes	1
20.19	odd	2		300.3.b.b.149.1		2
60.23	odd	4		300.3.g.a.101.1	✓	1
60.47	odd	4		300.3.g.c.101.1	yes	1
60.59	even	2		300.3.b.b.149.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.3.b.b.149.1		2	20.19	odd	2
300.3.b.b.149.1		2	60.59	even	2
300.3.b.b.149.2		2	4.3	odd	2
300.3.b.b.149.2		2	12.11	even	2
300.3.g.a.101.1	✓	1	20.3	even	4
300.3.g.a.101.1	✓	1	60.23	odd	4
300.3.g.c.101.1	yes	1	20.7	even	4
300.3.g.c.101.1	yes	1	60.47	odd	4
1200.3.c.b.449.1		2	1.1	even	1	trivial
1200.3.c.b.449.1		2	3.2	odd	2	CM
1200.3.c.b.449.2		2	5.4	even	2	inner
1200.3.c.b.449.2		2	15.14	odd	2	inner
1200.3.l.a.401.1		1	5.2	odd	4
1200.3.l.a.401.1		1	15.2	even	4
1200.3.l.e.401.1		1	5.3	odd	4
1200.3.l.e.401.1		1	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +13.0000i q^{7} -9.00000 q^{9} -23.0000i q^{13} +11.0000 q^{19} +39.0000 q^{21} +27.0000i q^{27} -59.0000 q^{31} +26.0000i q^{37} -69.0000 q^{39} +83.0000i q^{43} -120.000 q^{49} -33.0000i q^{57} -121.000 q^{61} -117.000i q^{63} +13.0000i q^{67} +46.0000i q^{73} -142.000 q^{79} +81.0000 q^{81} +299.000 q^{91} +177.000i q^{93} +167.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} + 22 q^{19} + 78 q^{21} - 118 q^{31} - 138 q^{39} - 240 q^{49} - 242 q^{61} - 284 q^{79} + 162 q^{81} + 598 q^{91}+O(q^{100}) \) 2 * q - 18 * q^9 + 22 * q^19 + 78 * q^21 - 118 * q^31 - 138 * q^39 - 240 * q^49 - 242 * q^61 - 284 * q^79 + 162 * q^81 + 598 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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